RD Sharma - Mathematics (Volume 1)

Book: RD Sharma - Mathematics (Volume 1)

Chapter: 9. Continuity

Subject: Maths - Class 12th

Q. No. 3 of Exercise 9.2

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3

Find the points of discontinuity, if any, of the following functions :

Basic Idea:


A real function f is said to be continuous at x = c, where c is any point in the domain of f if :


where h is a very small ‘+ve’ no.


i.e. left hand limit as x c (LHL) = right hand limit as x c (RHL) = value of function at x = c.


This is very precise, using our fundamental idea of limit from class 11 we can summarise it as, A function is continuous at x = c if :



Here we have,


…Equation 1


Function is defined for all real numbers so we need to comment about its continuity for all numbers in its domain ( domain = set of numbers for which f is defined )


Let c is any random number such that c ≠ 0 [thus c being random number, it is able to include all numbers except 0 ]


f(c) = [ using eqn 1]



Clearly,


We can say that f(x) is continuous for all x ≠ 0


As x = 0 is a point at which function is changing its nature so we need to check the continuity here.


Since, f(0) = 7 [using eqn 1]


NOTE : Idea of logarithmic limit and exponential limit –




You must have read such limits in class 11. You can verify these by expanding log(1+x) and ex in its taylor form.


Numerator and denominator conditions also hold for this limit like sandwich theorem.


E.g :


But,


and,



= [Using logarithmic and exponential limit as explained above, we have:]


=


Thus,


f(x) is discontinuous at x = 0


Hence, f is continuous for all x ≠ 0 but discontinuous at x = 0


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